Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications.

DefinitionEdit

If   and   are functors between the categories   and  , then a natural transformation   from   to   is a family of morphisms that satisfies two requirements.

  1. The natural transformation must associate, to every object   in  , a morphism   between objects of  . The morphism   is called the component of   at  .
  2. Components must be such that for every morphism   in   we have:
 

The last equation can conveniently be expressed by the commutative diagram

If both   and   are contravariant, the vertical arrows in this diagram are reversed. If   is a natural transformation from   to  , we also write   or  . This is also expressed by saying the family of morphisms   is natural in  .

If, for every object   in  , the morphism   is an isomorphism in  , then   is said to be a natural isomorphism (or sometimes natural equivalence or isomorphism of functors). Two functors   and   are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from   to  .

An infranatural transformation   from   to   is simply a family of morphisms  , for all   in  . Thus a natural transformation is an infranatural transformation for which   for every morphism  . The naturalizer of  , nat , is the largest subcategory of   containing all the objects of   on which   restricts to a natural transformation.

ExamplesEdit

Opposite groupEdit

Statements such as

"Every group is naturally isomorphic to its opposite group"

abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category   of all groups with group homomorphisms as morphisms. If   is a group, we define its opposite group   as follows:   is the same set as  , and the operation   is defined by  . All multiplications in   are thus "turned around". Forming the opposite group becomes a (covariant) functor from   to   if we define   for any group homomorphism  . Note that   is indeed a group homomorphism from   to  :

 

The content of the above statement is:

"The identity functor   is naturally isomorphic to the opposite functor  "

To prove this, we need to provide isomorphisms   for every group  , such that the above diagram commutes. Set  . The formulas   and   show that   is a group homomorphism with inverse  . To prove the naturality, we start with a group homomorphism   and show  , i.e.   for all   in  . This is true since   and every group homomorphism has the property  .

AbelianizationEdit

Given a group  , we can define its abelianization    . Let   denote the projection map onto the cosets of  . This homomorphism is "natural in  ", i.e., it defines a natural transformation, which we now check. Let   be a group. For any homomorphism  , we have that   is contained in the kernel of  , because any homomorphism into an abelian group kills the commutator subgroup. Then   factors through   as   for the unique homomorphism  . This makes   a functor and   a natural transformation, but not a natural isomorphism, from the identity functor to  .

Hurewicz homomorphismEdit

Functors and natural transformations abound in algebraic topology, with the Hurewicz homomorphisms serving as examples. For any pointed topological space   and positive integer   there exists a group homomorphism

 

from the  -th homotopy group of   to the  -th homology group of  . Both   and   are functors from the category Top* of pointed topological spaces to the category Grp of groups, and   is a natural transformation from   to  .

DeterminantEdit

Given commutative rings   and   with a ring homomorphism  , the respective groups of invertible   matrices   and   inherit a homomorphism which we denote by  , obtained by applying   to each matrix entry. Similarly,   restricts to a group homomorphism  , where   denotes the group of units of  . In fact,   and   are functors from the category of commutative rings   to  . The determinant on the group  , denoted by  , is a group homomorphism

 

which is natural in  : because the determinant is defined by the same formula for every ring,   holds. This makes the determinant a natural transformation from   to  .

Double dual of a vector spaceEdit

If   is a field, then for every vector space   over   we have a "natural" injective linear map   from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps are the components of a natural transformation from the identity functor to the double dual functor.

Finite calculusEdit

For every abelian group  , the set   of functions from the integers to the underlying set of   forms an abelian group   under pointwise addition. (Here   is the standard forgetful functor  .) Given an   morphism  , the map   given by left composing   with the elements of the former is itself a homomorphism of abelian groups; in this way we obtain a functor  . The finite difference operator   taking each function   to   is a map from   to itself, and the collection   of such maps gives a natural transformation  .

Tensor-hom adjunctionEdit

Consider the category   of abelian groups and group homomorphisms. For all abelian groups  ,   and   we have a group isomorphism

 .

These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors  . (Here "op" is the opposite category of  , not to be confused with the trivial opposite group functor on   !)

This is formally the tensor-hom adjunction, and is an archetypal example of a pair of adjoint functors. Natural transformations arise frequently in conjunction with adjoint functors, and indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit.

Unnatural isomorphismEdit

The notion of a natural transformation is categorical, and states (informally) that a particular map between functors can be done consistently over an entire category. Informally, a particular map (esp. an isomorphism) between individual objects (not entire categories) is referred to as a "natural isomorphism", meaning implicitly that it is actually defined on the entire category, and defines a natural transformation of functors; formalizing this intuition was a motivating factor in the development of category theory. Conversely, a particular map between particular objects may be called an unnatural isomorphism (or "this isomorphism is not natural") if the map cannot be extended to a natural transformation on the entire category. Given an object   a functor   (taking for simplicity the first functor to be the identity) and an isomorphism   proof of unnaturality is most easily shown by giving an automorphism   that does not commute with this isomorphism (so  ). More strongly, if one wishes to prove that   and   are not naturally isomorphic, without reference to a particular isomorphism, this requires showing that for any isomorphism  , there is some   with which it does not commute; in some cases a single automorphism   works for all candidate isomorphisms   while in other cases one must show how to construct a different   for each isomorphism. The maps of the category play a crucial role – any infranatural transform is natural if the only maps are the identity map, for instance.

This is similar (but more categorical) to concepts in group theory or module theory, where a given decomposition of an object into a direct sum is "not natural", or rather "not unique", as automorphisms exist that do not preserve the direct sum decomposition – see Structure theorem for finitely generated modules over a principal ideal domain § Uniqueness for example.

Some authors distinguish notationally, using   for a natural isomorphism and   for an unnatural isomorphism, reserving   for equality (usually equality of maps).

Example: fundamental group of torusEdit

As an example of the distinction between the functorial statement and individual objects, consider homotopy groups of a product space, specifically the fundamental group of the torus.

The homotopy groups of a product space are naturally the product of the homotopy groups of the components,   with the isomorphism given by projection onto the two factors, fundamentally because maps into a product space are exactly products of maps into the components – this is a functorial statement.

However, the torus (which is abstractly a product of two circles) has fundamental group isomorphic to  , but the splitting   is not natural. Note the use of  ,  , and  :[a]

 

This abstract isomorphism with a product is not natural, as some isomorphisms of   do not preserve the product: the self-homeomorphism of   (thought of as the quotient space  ) given by   (geometrically a Dehn twist about one of the generating curves) acts as this matrix on   (it's in the general linear group   of invertible integer matrices), which does not preserve the decomposition as a product because it is not diagonal. However, if one is given the torus as a product   – equivalently, given a decomposition of the space – then the splitting of the group follows from the general statement earlier. In categorical terms, the relevant category (preserving the structure of a product space) is "maps of product spaces, namely a pair of maps between the respective components".

Naturality is a categorical notion, and requires being very precise about exactly what data is given – the torus as a space that happens to be a product (in the category of spaces and continuous maps) is different from the torus presented as a product (in the category of products of two spaces and continuous maps between the respective components).

Example: dual of a finite-dimensional vector spaceEdit

Every finite-dimensional vector space is isomorphic to its dual space, but there may be many different isomorphisms between the two spaces. There is in general no natural isomorphism between a finite-dimensional vector space and its dual space.[1] However, related categories (with additional structure and restrictions on the maps) do have a natural isomorphism, as described below.

The dual space of a finite-dimensional vector space is again a finite-dimensional vector space of the same dimension, and these are thus isomorphic, since dimension is the only invariant of finite-dimensional vector spaces over a given field. However, in the absence of additional constraints (such as a requirement that maps preserve the chosen basis), the map from a space to its dual is not unique, and thus such an isomorphism requires a choice, and is "not natural". On the category of finite-dimensional vector spaces and linear maps, one can define an infranatural isomorphism from vector spaces to their dual by choosing an isomorphism for each space (say, by choosing a basis for every vector space and taking the corresponding isomorphism), but this will not define a natural transformation. Intuitively this is because it required a choice, rigorously because any such choice of isomorphisms will not commute with, say, the zero map; see (MacLane & Birkhoff 1999, §VI.4) for detailed discussion.

Starting from finite-dimensional vector spaces (as objects) and the identity and dual functors, one can define a natural isomorphism, but this requires first adding additional structure, then restricting the maps from "all linear maps" to "linear maps that respect this structure". Explicitly, for each vector space, require that it comes with the data of an isomorphism to its dual,  . In other words, take as objects vector spaces with a nondegenerate bilinear form  . This defines an infranatural isomorphism (isomorphism for each object). One then restricts the maps to only those maps   that commute with the isomorphisms:   or in other words, preserve the bilinear form:  . (These maps define the naturalizer of the isomorphisms.) The resulting category, with objects finite-dimensional vector spaces with a nondegenerate bilinear form, and maps linear transforms that respect the bilinear form, by construction has a natural isomorphism from the identity to the dual (each space has an isomorphism to its dual, and the maps in the category are required to commute). Viewed in this light, this construction (add transforms for each object, restrict maps to commute with these) is completely general, and does not depend on any particular properties of vector spaces.

In this category (finite-dimensional vector spaces with a nondegenerate bilinear form, maps linear transforms that respect the bilinear form), the dual of a map between vector spaces can be identified as a transpose. Often for reasons of geometric interest this is specialized to a subcategory, by requiring that the nondegenerate bilinear forms have additional properties, such as being symmetric (orthogonal matrices), symmetric and positive definite (inner product space), symmetric sesquilinear (Hermitian spaces), skew-symmetric and totally isotropic (symplectic vector space), etc. – in all these categories a vector space is naturally identified with its dual, by the nondegenerate bilinear form.

Operations with natural transformationsEdit

 
Horizontal and vertical composition of natural transformations

If   and   are natural transformations between functors  , then we can compose them to get a natural transformation  . This is done componentwise:  . This "vertical composition" of natural transformation is associative and has an identity, and allows one to consider the collection of all functors   itself as a category (see below under Functor categories).

Natural transformations also have a "horizontal composition". If   is a natural transformation between functors   and   is a natural transformation between functors  , then the composition of functors allows a composition of natural transformations  . This operation is also associative with identity, and the identity coincides with that for vertical composition. The two operations are related by an identity which exchanges vertical composition with horizontal composition.

If   is a natural transformation between functors  , and   is another functor, then we can form the natural transformation   by defining

 

If on the other hand   is a functor, the natural transformation   is defined by

 

Functor categoriesEdit

If   is any category and   is a small category, we can form the functor category   having as objects all functors from   to   and as morphisms the natural transformations between those functors. This forms a category since for any functor   there is an identity natural transformation   (which assigns to every object   the identity morphism on  ) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.

The isomorphisms in   are precisely the natural isomorphisms. That is, a natural transformation   is a natural isomorphism if and only if there exists a natural transformation   such that   and  .

The functor category   is especially useful if   arises from a directed graph. For instance, if   is the category of the directed graph • → •, then   has as objects the morphisms of  , and a morphism between   and   in   is a pair of morphisms   and   in   such that the "square commutes", i.e.  .

More generally, one can build the 2-category   whose

  • 0-cells (objects) are the small categories,
  • 1-cells (arrows) between two objects   and   are the functors from   to  ,
  • 2-cells between two 1-cells (functors)   and   are the natural transformations from   to  .

The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category   is then simply a hom-category in this category (smallness issues aside).

More examplesEdit

Every limit and colimit provides an example for a simple natural transformation, as a cone amounts to a natural transformation with the diagonal functor as domain. Indeed, if limits and colimits are defined directly in terms of their universal property, they are universal morphisms in a functor category.

Yoneda lemmaEdit

If   is an object of a locally small category  , then the assignment   defines a covariant functor  . This functor is called representable (more generally, a representable functor is any functor naturally isomorphic to this functor for an appropriate choice of  ). The natural transformations from a representable functor to an arbitrary functor   are completely known and easy to describe; this is the content of the Yoneda lemma.

Historical notesEdit

Saunders Mac Lane, one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations."[2] Just as the study of groups is not complete without a study of homomorphisms, so the study of categories is not complete without the study of functors. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations.

The context of Mac Lane's remark was the axiomatic theory of homology. Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly would be isomorphic to those of the singular theory. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.

See alsoEdit

NotesEdit

  1. ^ Zn could be defined as the n-fold product of Z, or as the product of Zn − 1 and Z, which are subtly different sets (though they can be naturally identified, which would be notated as ≅). Here we've fixed a definition, and in any case they coincide for n = 2.

ReferencesEdit

  • Mac Lane, Saunders (1998), Categories for the Working Mathematician, Graduate Texts in Mathematics 5 (2nd ed.), Springer-Verlag, p. 16, ISBN 0-387-98403-8
  • MacLane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), AMS Chelsea Publishing, ISBN 0-8218-1646-2.
  • Awodey, Steve (2010). Category theory. Oxford New York: Oxford University Press. p. 156. ISBN 978-0199237180.
  • Lane, Saunders (1992). Sheaves in geometry and logic : a first introduction to topos theory. New York: Springer-Verlag. p. 13. ISBN 0387977104.

External linksEdit